MSK modulation and differentially coherent detection transmission system

ABSTRACT

An embodiment of the transmission system comprises an MSK linear-modular transmitter and an MSK differentially coherent receiver connected by a transmission medium 3. A system embodying the invention is essentially characterized by filters in the receiver including analog or digital equalizers and by a differential encoding carried out on symbols of a message to be transmitted by a differential encoder included in the transmitter. The function of the equalizer is to eliminate intersymbol inteference in received signals. The differential encoding carried out is such that a coded symbol of rank k issued by the encoder depends on the corresponding symbol of the message and on the coded symbol of rank k-M, whereby M is an odd integer preferably greater than 1. On reception the probability of decision error with regard to a received symbol is all the lower that the integer M is high. The value M=3 is an appropriate value.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to continuous phase modulation in general. Moreparticularly, the invention relates to a binary modulation anddifferentially coherent detection transmission system.

The differentially coherent detection of a continuous phase signal is aparticularly interesting solution in situations where carrier recoveryis difficult, due to the simplicity of the receiver which is derivedfrom it and to its performances. The performances obtained for this typeof detection present a slight fall off by comparison with theperformances obtained with a coherent detection requiring a receiver ofgreater complexity.

Furthermore, continuous phase modulation with a modulation index equalto 0.5, known to those skilled in the art as MSK (Minimum Shift Keying)modulation, has the advantage of spectral compactness of the modulatedsignal notably by comparison with four-state phase modulation (PSK).

These two reasons, inter alia, result in MSK modulation withdifferentially coherent detection being considered for use notably fordigital data connections between mobile radio terminal equipment in theproject for a telecommunications by satellite system (MSAT) for mobileterminal equipment.

In this context, it is desirable to supply transmitter/receiverequipment providing increased performances by comparison with theexisting transmitter/receiver equipment.

2. Description of the Prior Art

M. K. SIMON and C. C. WANG describe in the article "DifferentialDetection of Gaussian MSK in a Mobile Radio Environment", IEEETransactions on Vehicular Technology, Vol. VT-33, No. 4, Nov. 1987, atransmission system with GMSK (Gaussian MSK) modulation anddifferentially coherent detection using in a receiver a Gaussian typeimpulse response receiver filter, in which a message comprised of astring of symbols to be transmitted from the transmitter towards areceiver is differentially coded in the transmitter into a string ofcoded symbols in such a way that a coded symbol of rank k in the stringdepends on the corresponding symbol of same rank in the message and onthe coded symbol of rank k-2. A delay of duration 2 T, where T is therate period at which the symbols are issued, is provided in the receiverso as to delay a received coded signal carrying the string of codedsymbols. The received signal and the delayed received signal aremultiplied so as to deduct by low-pass filtering from the productobtained a signal of which samples at instants kT are compared to anon-zero amplitude threshold so as to decide the values to be attributedto each of the message symbols.

This GMSK modulation transmission system reduces the probability ofdecision error with regard to the received symbols in comparison withother known equipment. A better aperture of the eye chart is obtainedsubsequent to the differential coding carried out according to which ak^(th) coded symbol depends on the (k-2)^(th) coded symbol and not onthe (k-1)^(th) as is usually the case. However, in this system, theintersymbol interference that is inherent to MSK modulation is noteliminated from the received signal. A greater reduction of theprobability of decision error can be obtained by a transmission systemin which the noise intersymbol interference would be eliminated from thesamples of the received signal.

OBJECT OF THE INVENTION

The object of this invention is to provide such a transmission system inwhich the intersymbol interference is eliminated from the samples of thereceived signal.

SUMMARY OF THE INVENTION

Accordingly, an MSK type continuous phase modulation transmission systemcomprises differentially coherent transmitting equipment and receivingequipment which are connected through a transmission medium. Thetransmitting equipment comprises means for coding an incoming messagecomprised of an ordered string of K symbols successively issued at apredetermined period, T, where K is any integer greater than one, intoan ordered coded string of K differentially coded symbols and deliversan MSK type continuous phase signal modulated by a string of impulsescarrying said ordered coded string. The receiving equipment comprisesadditional means in cascade with a matched filter for filtering insamples of a string of received impulses an intersymbol interferenceinherent to said MSK type continuous phase signal.

Said additional filtering means are comprised of an equalizer.

The receiver includes delay circuitry for multiplying a pair of receivedcomponents such that one of the components is delayed relative to theother components by MT.

The integer M is an odd integer thereby simplifying the structure of adecision circuit included in the receiving equipment. The theoreticalcalculations show that the probability of decision error is all thelower that the integer M is high. Nevertheless, almost all the possiblereduction of the probability of decision error through increase of theinteger M is obtained at M=3.

In the transmission system embodying the invention, for a sameprobability of decision error, the signal/noise ratio required for thereceived modulated signal in the receiving equipment is reduced byapproximately 1.5 dB by comparison with that required in thetransmission system described by M. K. SIMON and C. C. WANG.

BRIEF DESCRIPTION OF THE DRAWING

The foregoing and other objects, features and advantages of theinvention will be apparent from the following detailed description ofseveral embodiments of the invention with reference to the correspondingaccompanying drawings in which:

FIG. 1 is a simplified schematic diagram of the general structure of thetransmission system embodying the invention;

FIG. 2 is a schematic diagram of an analog or digital equalizer includedin a receiver of the system so as to eliminate the intersymbolinterference;

FIG. 3 is a block diagram of a first preferred embodiment of thetransmission system comprising a receiver with baseband receiverfilters;

FIG. 4 is a block diagram of a HILBERT type receiver with a band-passreceiver filter, included in a second preferred embodiment of thesystem;

FIG. 5 is a block diagram of a receiver of simplified structure of thesame type as the receiver in FIG. 5 deriving from the selection of acarrier frequency, of a rate frequency of the message symbols and of theinteger M such that their product is an integer; and

FIG. 6 is a block diagram of a transmitter with an MSK angle modulatorincluded in the transmission system embodying the invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

In order to facilitate the understanding of the functioning of thepractical preferred embodiments of the transmission system embodying theinvention, preferred embodiments described in reference to FIGS. 3 to 5,said system is first described, in reference to FIG. 1, for atransmission in baseband and in a simplified schematic formcorresponding to a complex model of an equivalent system in baseband. Ina complex space, the multiplication of a complex by j and -jrespectively corresponds to phase rotations of +π/2 and -π/2.Furthermore, an MSK signal being comprised by the sum of two carriers,cos (2πf₀ t) and -sin (2πf₀ t), phase shifted by -π/2 and respectivelymodulated by first and second differentially coded signals according tothe information to be transmitted, an MSK transmission system can berepresented in the form of a 2-transmission path system, a "real" firstpath modulating the carrier cos (2πf₀ t) and an "imaginary" second pathmodulating the carrier -sin (2πf₀ t). The MSK signal transmitted on atransmission medium towards a receiver is altered by the transmission.After demodulation in said receiver, the each of received first andsecond differentially coded signals contain part of the informationinitially carried entirely by the other coded signal. To a transmittedsymbol carried; e.g., by the first differentially coded signalcorresponds at reception a received sample represented in the complexplane by a real component obtained by sampling of the first signalreceived during a first time gap corresponding to the transmission ofthe symbols by the transmitter and an imaginary component obtained bysampling of the second signal received during the same first time gap.Likewise, to a transmitted symbol carried by the second differentiallycoded signal corresponds at reception a received symbol represented inthe complex plane by an imaginary component obtained by sampling of thesecond signal received during a second time gap corresponding to thetransmission of the symbol by the transmitter and a real componentobtained by sampling of the first signal received during the same secondtime gap.

From a mathematical analysis based on the theoretical model of thetransmission system embodying the invention are deducted the optimalstructure of a receiver filter included in a receiver of the system toproduce a signal from which is eliminated the intersymbol interference,as well as an optimal differential detection law that is to be carriedout by a decision circuit also included in said receiver so as to makethe decisions with regard to the values of symbols transmitted by atransmitter of the system, and with a minimal probability of error.

In reference to FIG. 1, the transmission system embodying the inventioncomprises an MSK transmitter 1, and an MSK differentially coherentreceiver 2. The transmitter 1 and the receiver 2 are connected by atransmission medium 3.

The transmitter is comprised of a differential encoder 11 and atransmitter filter 12.

The differential encoder 11 receives at input a message to betransmitted Me comprised of a string of K symbols α₀ to α_(K-1), where Kis an integer greater than one, the K symbols are mutually independent,with each taking on one of the equiprobable values 1, -1. At the encoder11 output is issued a string of K complex symbols a₀ to a_(K-1) inresponse to the input of symbols α₀ to α_(K-1), respectively. The codingcarried out by the encoder 11 being of the differential type, thesymbols a₀ to a_(K-1) are related by a differential coding function;this function is expressed by the equality:

    a.sub.k =j.α.sub.k.a.sub.k-M,                        (1)

where k is a whole index between 0 and K-1, M is a whole parameterpreferably odd and greater than 1, and j is the imaginary of unitymodule such that j² =-1. The symbols a₀ to a_(K-1) alternatively take onreal values 1 or -1 and imaginary values j or -j. In the known MSKtransmission systems, the parameter M usually takes on the value M=1,i.e. the value of a (k+1)^(th) symbol a_(k) issued by the differentialencoder depends on the value of the previous symbol a_(k-1) issued bythe differential encoder. In the system embodying the invention, thevalue of (k+1)^(th) symbol a_(k) issued by the encoder 11 depends on thevalue of the (k+1-M)^(th) symbol a_(k-M) issued by the encoder 11. Thefinality of this disposition will be more obvious at a later stage ofthe description. The encoder 11 memorizes initial symbols a_(-M) to a₋₁.The symbols a_(-M) to a₋₁ are determined in such a way that the symbolsa_(k) are equal to 1 or -1 when k is an even integer equal to 2n, andare equal to j or -j when k is an odd integer equal to 2n+1.

The transmitter filter 12 receives at its input the complex symbols a₀to a_(K-1) at a frequency rhythm 1/T and issues at its output an MSKsignal in baseband s_(b) (t). The complex envelope s_(b) (t) of the MSKsignal is expressed by the equality: ##EQU1## where E_(b) is the energyof the signal during the time gap of duration T, and h(t) is the impulseresponse of the filter 12 provided by the equalities:

    h(t)=(T).sup.-1/2.sin ((π/2T).t) for t between ε[0,2T]

    and

    h(t)=0, for t ε[0,2T].

The receiver 2 is comprised of a receiver filter 21, a sampler 22 ofsampling frequency 1/T, and a decision circuit 23.

The receiver filter 21 receives at its input an MSK signal in basebandz_(b) (t) corresponding to the transmitted signal s_(b) (t):

    z.sub.b (t)=s.sub.b (t).exp (jθ)+n(t),               (3)

where exp (jθ) and n(t) respectively represent a phase shift and acomplex Gaussian noise introduced by the transmission of the signals_(b) (t) through the transmission medium 3. At its output, the filter21 supplies a signal y(t) expressed by the convolution product: ##EQU2##where g(-t) represents the impulse response of the receiver filter 21and τ a temporal variable.

The signal y(t) is sampled by the sampler 22 at frequency 1/T. At thesampling instants kT, the sampler 22 supplies samples y(kT) of thesignal y(t), also noted y_(k). From the equalities (2), (3) and (4), itensues that a sample y_(k) is determined by the equality: ##EQU3## wherem is a whole number index and V_(k) is a noise caused by the integrationof the noise n(t).

In a system embodying the invention, the samples y_(k) must be freedfrom the intersymbol interference so as to reduce the probability ofdecision error. A sample y_(k) must only depend on the symbol a_(m) 32 kto the exclusion of another symbol a_(m) ≠k; this condition is onlyfulfilled if the integral: ##EQU4## is zero for all m≠k; which amountsto satisfying Nyquist's condition: ##EQU5## whereby δ₀,k representsKronecker's symbol equal to 1 for k=0 and equal to 0 for k≠0.

The sample y_(k) is then expressed by the equality:

    y.sub.k =(2E.sub.b).sup.1/2.exp (jθ).a.sub.k +V.sub.k.(7)

In order to preserve the optimity of the filter 21, the function h(t)must necessarily be decomposable into a sum of functions g(t-qT). Thefollowing equality must be verified: ##EQU6## whereby the coefficientsp_(q) are real coefficients respectively associated with the functionsg(t-qT) and the index q is an integer that takes on positive andnegative values between -Q and Q. In the complex plane of Fouriertransforms, the equality (8) becomes: ##EQU7## where H(f) and G(f) arerespectively the Fourier transforms of the functions h(t) and g(t). Fromthis last equality is deducted the expression of the transfer functionG*(f) of the receiver filter 21, which transfer function is the Fouriertransform of the function g(-t). The expression: ##EQU8## being real,the function G*(f) is determined by the equality: ##EQU9## where thesymbol * indicates the corresponding conjugate complex.

From the expression (9) is derived the receiver filter 21 which iscomprised of a conventional filter 21 matched with the waveform of theimpulse response h(t) and having a transfer function H*(f) which is theconjugate complex of the transfer function H(f) of the transmitterfilter 12, and of an equalizer 212 having a transfer function E_(r) (f)approximating the theoretical transfer function: ##EQU10##

The coefficients p_(q) are calculated from the expression: ##EQU11##This expression is deducted from the equality (8) by calculation of theconvolution product on each side of the equality (8) and by applicationof Nyquist's condition (6). The calculation of the coefficients p_(q)for the expression h(t) given after the equality (2), gives:

    p.sub.-1 =1/π

    p.sub.0 =1

    p.sub.1 =1/π

    and

    p.sub.q =0 for q ε[-1,0,1]

Given E(z), the z-transform corresponding to the function E(f):

    E(z)=1/(1+(1/π).z.sup.-1 +(1/π).z)

The transform E(z) of the transfer function E(f) comprises an unstablepole, i.e. of module greater than 1, and cannot therefore be achievedpractically. However, the theoretical transfer function E(f) isapproximated by the transfer function E_(r) (f) which can be achievedpractically and calculated to minimize, in the absence of noise, the RMSerror between the responses due at E(f) and E_(r) (f). The transferfunction E_(r) (f) must be shown to satisfy the expression: ##EQU12## inwhich the coefficients c_(q) are such that c_(q) =c_(-q) and satisfy thematrix relation:

    P'=B×C,

where P' and C are vectors having respectively as components thecoefficients p_(-q) and c_(q), and B is a square matrix having (2Q+1)²coefficients such that: ##EQU13## i and j respectively representing theline and the column in the matrix B of the coefficient b_(ij) inquestion.

The equalizer 212 can be realized in the form of an analog equalizer bymeans of delay circuits comprised of delay lines. It can also berealized in digital form, e.g. by means of a micro-computer or a digitalcircuit. In the latter case, it comprises at its input a samplerfollowed by an analog/digital converter. In such an instance the sampler22 of the receiver 2 is not included and it is possible to realize thefunction of the decision circuit 23 by said micro-computer. Furthermore,the equalizer 212 and the matched filter 211 can be realized in the formof a single digital or analog filter 21.

The general structure of the equalizer 212 is shown in FIG. 2. Thisstructure is conventional and is directly derived from the expression ofE_(r) (f). The equalizer 212 comprises 2Q delay circuits 2120.sub.(-Q+1)to 2120_(Q), 2Q+1 amplifiers 2121_(-Q) to 2121_(Q), and a 2Q+1-inputsumming integrator or adder 2122. The delay circuits each realize adelay of duration equal to T. The summing circuit 2122, whichapproximates an integrator, issues at its output the signal y(t) freedfrom the intersymbol interference at the sampling instants kT.

Though freed from the intersymbol interference by the filter 21, thesamples y_(k) contain a noise V_(k), as shown by the equality (7). Thenoise V_(k) is a correlated noise; its auto-correlation function isequal to:

    A(V.sub.k, V.sub.k-u)=2N.sub.0.q.sub.u,

whereby u is a whole index, 2N₀ is the power spectral density of thenoise n(t) in the frequency band of the signal s_(b) (t) transmitted bythe transmitter 1, and where q_(u) is the coefficient of index u givenby the equality: ##EQU14##

The decision circuit 23 is designed to maximize the expression:##EQU15## where Re[.] is the real component of the complex productbetween square brackets, a^(i) _(k) and a^(i*) _(k-M) being respectivelythe complex symbols liable to be transmitted, where I₀ [.] is themodified Bessel function of degree zero, and |Y^(*t).R⁻¹ _(v).A_(I) | isthe module of the scalar Y^(*t).R⁻¹ _(v).A_(I), Y and A_(I) beingrespectively vectors of coordinates (y_(k), y_(k-M)) and (a^(i) _(k),a^(i) _(k-M)), R⁻¹ _(v) being the inverse matrix to a noise correlationmatrix: ##EQU16## the coefficients q_(u) being determined by theequality (10), and Y^(*t) being the transposed line vector correspondingto column vector Y*.

The decision circuit 23 must make the decision that maximizes theexpression (11). To simplify the expression (11) M is odd. The producta^(i) _(k).a^(i*) _(k-M) is then an imaginary, given that thecoefficients a^(i) _(k) are alternatively real and imaginary. Bydeveloping the expression (11), the expression of the optimal decisionα_(k) corresponding to the symbol α_(k) can be found:

    α.sub.k =1. sign of [Im[y.sub.k.y*.sub.k-M ]].       (12)

It can also be shown that the probability of decision error made on asymbol α_(k) decreases as M increases. For M having an infinite value inpractice greater than or equal to 3, the probability of decision erroris equal to: (1/2) exp (-E_(b) /N₀.q₀). This error in the case of anequivalent transmission system with binary differential modulation attwo phase states (BSPK) taken as a reference is given by the knownexpression:

    (1/2).exp (-E.sub.b /N.sub.0).

For the same probability of decision error, a transmission systemembodying the invention must therefore have a signal/noise ratio equalto E_(b) /N₀, greater in dB by 10 log q₀ by comparison with thesignal/noise ratio required for the BPSK modulation transmission system.For the coefficients p₀ =1, p₁ =p₋₁ =1/π, the coefficient q₀ is equal to1.3, whence:

    10 log q.sub.0 =1.13 dB.

Such a result represents an improvement of at least 1.5 dB compared tothe state of the art notably compared to the transmission systemdescribed by M. K. SIMON and C. C. WANG.

A first preferred embodiment of the transmission system embodying theinvention comprising an MSK transmitter 1a and a receiver 2a withbaseband receiver filters is shown in FIG. 3.

The MSK transmitter 1a is of the linear modulator type. It comprises adifferential encoder 11, and a linear modulator 13.

The differential encoder 11 receives at its input the message to betransmitted Me comprised of K symbols α₀ to α_(K-1) and issues thecomponents Re[a_(2n) ] of the real symbols a_(2n) and the componentsIm[a_(2n) +1] of the imaginary symbols a_(2n+1) at first and secondoutputs 111 and 112. The differential encoder 11 is a sequential logiccircuit easily realized by those skilled in the art by means of logicgates and sweep circuits; its internal structure is therefore notdescribed in detail herein.

The linear modulator 13 comprises 2 transmitter filters 12r and 12i, twoanalog multiplying devices 131r and 131i and an analog summingintegrator 132.

The transmitter filters 12r and 12i are analogous and have both theimpulse response h(t) defined previously. The filter 12r receives at itsinput the components Re[a_(2n) ]=a_(2n) of the symbols a_(2n) with aneven index and issues at its output a signal comprised of a string ofamplitude modulated impulses amplitude s_(r) (t). The filter 12ireceives at its input the components Im[a_(2n+1) ]=a_(2n+1) /j of thesymbols a_(2n+1) with an odd index and issues at its output a signalcomprised of a string of amplitude modulated impulses amplitude s_(i)(t).

The multiplying device 131r receives the signal s_(r) (t) and asinusoidal carrier cos (2πf₀ t) of frequency f₀ at first and secondinputs respectively and issues at its output on amplitude modulatedcarrier frequency signal s_(r) (t). cos (2πf₀ t). The multiplying device131i receives the signal s_(i) (t) and a sinusoidal carrier -sin (2πf₀t) of frequency f₀ and phase shifted by -π/2 by comparison with thecarrier cos (2πf₀ t), at first and second inputs respectively. Themultiplying device 131i issues at its output an amplitude modulatedcarrier frequency signal -s_(i) (t).sin (2πf₀ t). The summing integrator132 receives the signals s_(r) (t).cos (2πf₀ t) and -s_(i) (t).sin (2πf₀t) at first and second inputs respectively and issues at its output anMSK signal at carrier frequency s(t)=s_(r) (t).cos (2πf₀ t)- s_(i)(t).sin (2πf₀ t). The signal s(t) is transmitted to the receiver 2athrough the transmission connection 3.

The receiver 2a comprises a demodulator 20, two receiver filters 21r and21i, two samplers 22r and 22i, and a decision circuit 23.

The demodulator 20 comprises two analog multiplying devices 201r and201i and a non-triggered local oscillator (not shown) issuing twosinusoidal carriers, 2 cos (2πf₀ t) and -2 sin (2πf₀ t), of nominalfrequency f₀ and phase shifted from one another by -π/2.

The multiplying devices 201r and 201i are analogous and both receiverespectively at first inputs a signal z(t) corresponding to a signals(t) phase shifted and noised by its transmission through the connection3. The multiplying device 201r receives the sinusoidal carrier offrequency f₀, 2 cos (2πf₀ t), at a second input and issues at output thesignal 2 z(t).cos (2πf₀ t) comprised of a sum of a real component of thesignal s_(r) (t), of a real component of the signal s_(i) (t), and ofother components around the frequency 2f₀ in the frequency spectrum. Themultiplying device 201i receives the sinusoidal carrier -2 sin (2πf₀ t)of frequency f₀ at a second its input and issues at output the signal-2.z(t).sin (2πf₀ t) comprised of a sum of imaginary components of thesignal s_(i) (t) of an imaginary component of the signal s_(r) (t), andof other components around the frequency 2f₀ in the frequency spectrum.

The receiver filters 21r and 21i are analogous to the filter 21described in reference to FIG. 1. They are both comprised of a matchedfilter and of an equalizer. The filter 21r receives at its input thesignal 2.z(t).cos (2πf₀ t) and issues at its output a signal y_(r) (t)representing the real components of signals s_(r) (t) and s_(i) (t)included in the signal 2.z(t).cos (2πf₀ t). The other components aroundthe frequency 2f₀ included in the signal 2z(t).cos (2f₀ t) areeliminated by the low-pass filter 21r. The filter 21i receives at itsinput the signal -2.z(t).sin (2πf₀ t) and issues at its output a signaly_(i) (t) representing the imaginary components of the signals s_(i) (t)and s_(r) (t) included in the signal -2.z(t).sin (2πf₀ t). The othercomponents around the frequency 2f₀ included in the signal -2.z(t).sin(2f₀ t) are eliminated by the filter 21i. The signals y_(r) (t) andy_(i) (t) represent respectively the real and imaginary components ofthe complex signal y(t) (FIG. 1) corresponding to the real symbolsa_(2n) and the imaginary symbols a_(2n+1) transmitted.

The samplers 22r and 22i are analogous. They receive at their inputs thesignals y_(r) (t) and y_(i) (t) respectively and issue correspondingsamples y_(rk) and y_(ik) at the sampling instants kT. The samplesy_(rk) and y_(ik) are respectively the real and imaginary components ofa complex sample y_(k) =y_(rk) +j.y_(ik) corresponding to a complexsymbol a_(k).

The decision circuit 23 comprises a Im[y_(k).y*_(k-M) ] calculatingcircuit 231, and a sign detector 232.

The Im[y_(k).y*_(k-M) ] calculating circuit 231 comprises two delaycircuits 2311r and 2311i, two analog multiplying devices 2312a and 2312band an analog summing integrator 2313. The circuit 231 receives thesamples y_(rk) and y_(ik) issued by the samplers 22r and 22i.

The sample y_(rk) is applied in parallel to the input of the delaycircuit 2311r and a first input of the multiplying device 2312a. Thesample y_(ik) is applied in parallel to the input of the delay circuit2311i and a first input of the multiplying device 2312b. The delaycircuits 2311r and 2311i respectively issue samples y_(r)(k-M) andy_(i)(k-M) at outputs thereof. The samples y_(r)(k-M) and y_(i)(k-M) arerespectively the real and imaginary components of the complex sampley_(k-M) corresponding to the complex symbol a_(k-M). The multiplyingdevices 2312a and 2312b issue at outputs thereof the productsy_(rk).y_(i)(k-M) and y_(ik).y_(r)(k-M). The product signals aresupplied respectively to inverting (-), and non-inverting (+) inputs ofthe summing integrator 2313. The summing integrator 2313 supplies at itsoutput the imaginary component Im[y_(k).y*_(k-M) ]=y_(ik).y_(r)(k-M)y_(rk).y_(i)(k-M) of the complex product y_(k).y_(k-M).

The sign detector 232 receives at its input the imaginary componentIm[y_(k).y*_(k-M) ] supplied by the calculating circuit 231 and issuesat its output a decision α_(k) =1 when said component is positive, andequal to α_(k) =-1 when it is negative.

A second preferred embodiment of the transmission system embodying theinvention comprises a transmitter analogous to the transmitter 1a shownin FIG. 3 and a receiver 2b, FIG. 4, of the HILBERT receiver type with aπ/2 phase shift, which receiver does not require a local oscillator fordemodulation of the received signal z(t).

The rear of FIG. 4 comprises a band-pass receiver filter 24 having amidband frequency equal to the carrier frequency f₀, a demodulator 25,and a decision circuit 27.

The band-pass receiver filter 24 has an impulse response c(t) and is theband-pass filter of midband frequency f₀ corresponding to the basebandreceiver filter 21r, 21i shown in FIG. 3. The impulse response c(t) ofthe band-pass filter 24 is deducted, as is known, from the impulseresponse g(-t) of the baseband filter 21r, 21i. The impulse responsec(t) of the filter 24 is given by the expression:

    c(t)=2.Re[g(-t).exp (j2πf.sub.0 t)].

The structure of the filter 24 remains analogous to that of the basebandfilter 22r, 22i. The filter 24 comprises a matched band-pass filter andan equalizer.

The receiver filter 24 receives at its input the signal z(t) and issuesat its output a signal y_(t) (t): ##EQU17## where y(t) is the complexsignal previously defined having real y_(r) (t) and imaginary y_(i) (t)components, respectively derived by the corresponding baseband filters21r and 21i in response to the demodulated signal z(t) (FIG. 3). Thesignal y_(t) (t) is supplied to the demodulator 25.

The demodulator 25 comprises two analog multiplying devices 251a and251b, a delay circuit 252, a -π/2 phase converter 253, and two low-passfilters 254a and 254b.

The signal y(t) is applied is parallel to the input of the delay circuit252 and to a first input of each of multiplying devices 251a and 251b.The delay circuit supplies at its output a signal y_(t) (t-MT)corresponding to the signal y_(t) (t) delayed by a duration equal to MT:

    y.sub.t (t-MT)=y.sub.r (t-MT).cos (2πf.sub.0 (t-MT))-y.sub.i (t-MT).sin (2πf.sub.0 (t-MT)).

The signal y_(t) (t-MT) is applied is parallel to the input of the phaseconverter 253 and a second input of the multiplying device 251a.

The phase converter 253 shifts by -π/2 the phase of the carrier offrequency f₀ and does not alter the other components of the signal y_(t)(t-MT) of frequency different to f₀. The phase converter 253 issues atits output a signal y_(t) (t-MT)₋π/2 corresponding to the signal y_(t)(t-MT) which is phase shifted by -π/2:

    y.sub.t (t-MT).sub.-π/2 =y.sub.r (t-MT).sin (2πf.sub.0 (t-MT))+y.sub.i (t-MT).cos (2πf.sub.0 (t-MT)).

The phase converter 253 supplies the signal y_(t) (t-MT)₋π/2 at a secondinput of the multiplying device 251b.

The multiplying devices 251a and 251b respectively issue at outputsthereof the products:

    y.sub.t (t).y.sub.t (t-MT) and y.sub.t (t).y.sub.t (t-MT).sub.-π/2.

The products y_(t) (t).y_(t) (t-MT) and y_(t) (t).y_(t) (t-MT)₋π/2 arerespectively applied to inputs of the low-pass filters 254a and 254b.

The function of the low-pass filters 254a and 254b is to eliminate thecomponents of the products y_(t) (t).y_(t) (t-MT) and y_(t) (t).y_(t)(t-MT)₋π/2 around the frequency 2f₀. The low-pass filters 254a and 254brespectively issue signals y² _(a) (t) and y² _(b) (t) corresponding tothe products y_(t) (t).y_(t) (t-MT) and y_(t).y_(t) (t-MT)₋π/2 afterlow-pass filtering: ##EQU18##

The signals y² _(a) (t) and y² _(b) (t) are respectively supplied toinputs of the samplers 26a and 26b which issue outputs thereof samplesy² _(ak) and y² _(bk). The samples y² _(ak) and y² _(bk) are supplied tothe decision circuit 27.

The decision circuit 27 comprises an Im[y_(k).y*_(k-M) ] calculatingcircuit 271 and a sign detector 272.

The calculating circuit 271 comprises two analog multiplying devices2711a and 2711b and an analog summing integrator 2712. The multiplyingdevices 2711a and 2711b are analogous and respectively receive at firstinputs thereof the samples y² _(ak) and y² _(bk). The multiplyingdevices 2711a and 2711b also respectively receive at second inputsthereof direct voltages of amplitude -sin (2πf₀ MT) and cos (2πf₀ MT).The multiplying device 2711a issues at a first output of the summingcircuit 2712 the product -y² _(ak).sin (2πf₀ MT); the multiplying device2711b issues at a second output of the summing circuit 2712 the producty² _(bk).cos (2πf₀ MT). The summing circuit 2712 issues at its outputthe sum of the products -y² _(ak).sin (2πf₀ MT) and y² _(bk).cos (2πf₀MT) equal to:

    (1/2).[y.sub.ik.y.sub.r(k-M) -y.sub.rk.y.sub.i(k-M) ]=(1/2).Im[y.sub.k.y*.sub.k-M ].

The sign detector 272 is analogous to the sign detector 232 shown inFIG. 3. Detector 272 receives at its input the signal 2.Im[_(k).y*_(k-M)] issued by the summing circuit 2712, detects the sign of the signal andsupplies at its output a decision α_(k) =1 if the detected signal ispositive and α_(k) =-1 if the detected signal is negative.

A third preferred embodiment of the transmission system embodying theinvention comprises a transmitter analogous to the transmitter 1a shownin FIG. 3 and a receiver 2c, FIG. 5, having a simplified structure incomparison with the receivers 2a and 2b shown in FIGS. 3 and 4.

The expression y² _(b)(t) given previously shows that it is possible tosimplify the structure of the receiver 2b by selecting the frequency f₀and the duration M.T in such a way that the product f₀.M.T is aninteger. Indeed, in this case sin (2πf₀ MT)=0 and cos (2πf₀ MT)=1 andthe signal y² _(b) (t) is then equal to (1/2)[y_(i) (t).y_(r)(t-MT)-y_(r) (t).y_(i) (t-MT)]. The signal (1/2)Im[y_(k).y*_(k-M)]=(1/2)[y_(ik).y_(r)(k-M) -y_(rk).y_(i)(k-M) ] can thus be directlyobtained by sampling the signal y² _(b) (t) at the instants kT. Thereceiver 2c, shown in FIG. 5, is designed to function in the event ofthe product f₀.M.T being an integer.

The receiver 2c is obtained by removing from the receiver 2a all thecircuits intended to produce the signal y² _(a) (t), and the signal(1/2)Im[y_(k).y*_(k-M) ] by combining the signals y² _(a) (t) and y²_(b) (t).

The receiver 2c therefore comprises the band-pass receiver filter 24, ademodulator 25c derived from the demodulator 25 by eliminating themultiplying device 251a and the low-pass filter 254a, the sampler 26band the sign detector 272. The samples y² _(b) (t) issued by the sampler26b are directly applied at input of the sign detector 272.

According to another preferred embodiment of the transmission systemembodying the invention, the MSK linear-modulator transmitter 1a shownin FIG. 3 is replaced by an MSK angle-modulator transmitter 1b shown inFIG. 6.

The MSK angle-modulator transmitter 1b is comprised of a differentialencoder 11b, of a filter 12b with a rectangular impulse response hb(t)of duration T and of a voltage controlled oscillator 13b (VCO) of middlefrequency f₀, these three circuits being connected in cascade and in theabove-mentioned order.

The differential encoder 11 receives at its input the message to betransmitted Me comprised of the symbols α₀ to α_(K-1) and issues at itsoutput a string of coded symbols β₀ to β_(K-1). The symbols β₀ toβ_(K-1) are respectively issued in response to the symbols α₀ toα_(K-1). The encoder 11b carries out a differential coding functionexpressed by the equality: ##EQU19##

The symbols β₀ to β_(K-1) are supplied at input of the filter 12b andamplitude modulate rectangular impulses of duration T derived by therectangular-impulse-response filter 12b. These modulated rectangularimpulses constitute a signal comprised of a string of impulses s_(f) (t)applied to a modulation input of the VCO 13b. The signal s_(f) (t)frequency modulates with a modulation index m=0.5 a carrier of frequencyf₀ supplied by the VCO 13b. The VCO issues a signal s₀ (t) modulated infrequency at index m=0.5, equivalent to an MSK signal.

What is claimed is:
 1. A minimum shift key (MSK) continuous phasemodulation transmission system comprising differentially coherent typeMSK transmitting equipment and differentially coherent type MSKreceiving equipment which are connected through a transmissionmedium,said transmitting equipment comprising means for encoding amessage having an ordered string of K symbols successively derived witha predetermined periodicity, T, into an ordered coded string of Kdifferentially coded symbols, where K is an integer greater than one,the encoder deriving an MSK continuous phase signal modulated by awaveform having a string of impulses carrying said ordered coded string,said receiving equipment including: filter means matched with thewaveform of the impulses of said impulse string, and means cascaded withsaid filter means for suppressing intersymbol interference inherent tosaid MSK type continuous phase signal from samples of a string ofreceived impulses.
 2. The transmission system claimed in claim 1,wherein said means for suppressing comprises an analog or digitalequalizer.
 3. The transmission system claimed in claim 1, wherein saidencoding means performs a differential coding function such that anycoded symbol of rank k in said coded string depends on the product ofthe coded symbols of ranks k-1 to k-M+1 and on the symbol of rank k insaid incoming message, where k is successively every integer from 1 toK, and M is a delay time having an integral value greater than one. 4.The transmission system of claim 1 wherein the signal received by thereceiver has a frequency f₀ and the means for suppressing includes meansfor deriving a first signal proportional to the product of the receivedsignal and the received signal delayed by the linear combination of MTand one quarter of a cycle at f₀, where f₀ MT and M have integral valuesso sin 2πf₀ MT=0 and cos 2πf₀ MT=1, and means for periodically samplingthe first signal at times spaced from each other by T.
 5. Thetransmission system of claim 4 further including means for detecting thepolarity of the sampled first signal.
 6. The transmission system ofclaim 1 wherein the means for suppressing includes first and secondchannels respectively responsive to the signal received by the receiverfor respectively deriving first and second signals having orthogonallyrelated components, means for simultaneously and periodically samplingthe first and second signals at times spaced from each other by T, andmeans for combining the sampled first and second signals.
 7. Thetransmission system of claim 6 wherein the first channel includes afirst multiplier responsive to the received signal and a first referencewave having a first phase and a frequency equal to a carrier of thereceived signal and a first matched filter responsive to an output ofthe first multiplier, the second channel including a second multiplierresponsive to the received signal and a second reference wave having thecarrier frequency and a second phase orthogonal to the first phase and asecond matched filter responsive to an output of the second multiplier,the first and second matched filters supplying signals to the samplingmeans, the first and second matched filters being included in saidfilter means.
 8. The transmission system of claim 7 wherein thecombining means includes a third multiplier responsive to the sampledsignal derived by the first channel and a replica of the sampled signalderived by the second channel delayed by MT, where M is an integer, anda fourth multiplier responsive to the sampled signal derived by thesecond channel and a replica of the sampled signal derived by the firstchannel delayed by MT, and means for linearly combining product signalsderived by the third and fourth multipliers.
 9. The transmission systemof claim 8 further including means for detecting the polarity of anoutput signal of the means for linearly combining.
 10. The transmissionsystem of claim 9 wherein M is an odd integer greater than one.
 11. Thetransmission system of claim 5 wherein M is an odd integer greater thanone.
 12. The transmission system of claim 6 wherein the filter means isresponsive to the signal received by the receiver, the first channelincluding a first multiplier responsive to an output signal of thefilter means and the output signal as delayed by MT, where M is aninteger, the second channel including a second multiplier responsive tothe output signal and the output signal as delayed by a linearcombination of MT and one quarter cycle of the output signal, thesampling means being responsive to orthogonal components derived fromsaid first and second multipliers.
 13. The transmission system of claim12 wherein the means for combining includes: a third multiplierresponsive to a sampled signal derived by said sampling means inresponse to the first multiplier and a first reference wave having thefrequency f₀ MT and a reference phase, where f₀ is the frequency derivedby said filter means, a fourth multiplier responsive to a sampled signalderived by said sampling means in response to the second multiplier anda second reference wave having the frequency f₀ MT and a phaseorthogonally related to the reference phase, and means for linearlycombining components derived from said third and fourth multipliers. 14.The transmission system of claim 13 further including means fordetecting the polarity of an output signal of the means for linearlycombining.
 15. An MSK type continuous phase modulation transmissionsystem comprising differentially coherent type transmitting equipmentand differentially coherent type receiving equipment which are connectedthrough a transmission medium, wherein said transmitting equipment is ofthe linear modulator type and comprises means for encoding an incomingmessage including an ordered string of K symbols successively derived ata predetermined period into an ordered coded string of K differentiallycoded symbols, where K is any integer greater than one, said encodingmeans deriving an MSK type continuous phase signal modulated by a stringof impulses carrying said ordered coded string, said encoding meanscarrying out a differential coding function such that any coded symbolof rank k in said coded string depends on the product of the codedsymbol of rank k-M and the symbol of rank k in said incoming message,where M is a predetermined odd integer greater than 1,saiddifferentially coherent type receiving equipment comprising a filtermatched with the waveform of the impulses of said impulse string, andmeans cascaded with said filter for suppressing intersymbol interferenceinherent to said MSK type continuous phase signal from samples of astring of received impulses.
 16. A transmission system as claimed inclaim 15, wherein said period at which are issued the K symbols of saidmessage, the odd integer M, and a carrier frequency of said MSK typecontinuous phase signal are predetermined in such a way that theirproduct is an integer.
 17. An MSK type continuous phase modulationtransmission system comprising differentially coherent type transmittingequipment and receiving equipment which are connected through atransmission medium, whereinsaid transmitting equipment comprises meansfor encoding a message having an ordered string of K symbolssuccessively derived at a predetermined period into an ordered codedstring of K differentially coded symbols and for deriving an MSKcontinuous phase signal modulated by a string of impulses carrying saidordered coded string, where K is any integer greater than one, saidreceiving equipment comprising: filter means matched with the waveformof the impulses of said impulse string, and means cascaded with saidfilter means for suppressing intersymbol interference inherent to MSKtype continuous phase signals, and thereby deriving first and secondsignals comprising said received impulse string, wherein said receivingequipment further comprises means for deciding with a minimalprobability of error the value of a k^(th) symbol in said messageaccording to first and second values of said first and second signalsincluding said received impulse string respectively at first and secondsampling instants, said first and second sampling instants being ofranks k-M and k respectively corresponding to the transmission by saidtransmitting equipment of said k-M^(th) and k^(th) coded symbols, whereM is an odd integer greater than 1, said first values of said first andsecond signals at the first sampling instant of rank k-M respectivelyrepresenting first and second components of a k-M^(th) received codedsymbol corresponding to the k-M^(th) symbol of said message, said secondvalues of said first and second signals at said second sampling instantof rank k respectively representing first and second components of ak^(th) received coded symbol corresponding to the k^(th) symbol of saidmessage.
 18. A transmission system as claimed in claim 17, wherein saiddeciding means comprises means for calculating the difference betweenthe product of the first component of said k-M^(th) received codedsymbol and of said first component of said k^(th) received coded symbol,and means for detecting the sign of said difference so as to decidedirectly from said sign the value to be attributed to said k^(th) symbolof said message.
 19. Receiving equipment in an MSK transmission systemwherein an encoder at a transmitter derives a message having an orderedstring of K symbols successively derived at a predetermined period Tinto an ordered coded string of K differentially coded symbols and forderiving an MSK continuous phase signal modulated by a string ofimpulses carrying said ordered coded string, where K is any integergreater than one,said receiving equipment comprising a filter meansmatched with the waveform of the impulses of said impulse string, andmeans cascaded with said filter means for suppressing intersymbolinterference inherent to said MSK type continuous phase signals, andthereby deriving first and second signals comprising said receivedimpulse string, wherein said receiving equipment further comprises meansfor deciding with a minimal probability of error the value of a k^(th)symbol in said message according to first and second values of saidfirst and second signals including said received impulse stringrespectively at first and second sampling instants, said first andsecond sampling instants being of ranks k-M and k respectivelycorresponding to the transmission by said transmitting equipment of saidk-M^(th) and k^(th) coded symbols, where M is an odd integer greaterthan 1, said first values of said first and second signals at the firstsampling instant of rank k-M respectively representing first and secondcomponents of a k-M^(th) received coded symbol corresponding to thek-M^(th) symbol of said message, said second values of said first andsecond signals at said second sampling instant of rank k respectivelyrepresenting first and second components of a k^(th) received codedsymbol corresponding to the k^(th) symbol of said message.
 20. Thereceiver equipment of claim 19, wherein said deciding means comprisesmeans for calculating the difference between the product of the firstcomponent of said k-M^(th) received coded symbol and of said firstcomponent of said k^(th) received coded symbol, and means for detectingthe sign of said difference so as to decide directly from said sign thevalue to be attributed to said k^(th) symbol of said message. 21.Receiving equipment in an MSK transmission system wherein an encoder ata transmitter derives a message having an ordered string of K symbolssuccessively derived with a predetermined periodicity, T, into anordered coded string of K differentially coded symbols, where K is aninteger greater than one, the encoder deriving an MSK continuous phasesignal modulated by a waveform having a string of impulses carrying saidordered coded string, said receiving equipment comprising filter meansmatched with the waveform of the impulses of said impulse string, andmeans cascaded with said filter means for suppressing intersymbolinterference inherent to said MSK type continuous phase signal fromsamples of a string of received impulses, the signal received by thereceiving equipment having a frequency f₀ and the means for suppressingincludes means for deriving a first signal proportional to the productof the received signal and the received signal delayed by the linearcombination of MT and one quarter of a cycle at f₀, where f₀ MT and Mhave integral values so sin 2πf₀ MT=0 and cos 2πf₀ MT=1, and means forperiodically sampling the first signal at times spaced from each otherby T.
 22. The receiving equipment of claim 21 further including meansfor detecting the polarity of the sampled first signal.
 23. Thereceiving equipment of claim 22 wherein the means for suppressingincludes first and second channels respectively responsive to the signalreceived by the receiver for respectively deriving first and secondsignals having orthogonally related components, means for simultaneouslyand periodically sampling the first and second signals at times spacedfrom each other by T, and means for combining the sampled first andsecond signals.
 24. The receiving equipment of claim 23 wherein thefirst channel includes a first multiplier responsive to the receivedsignal and a first reference wave having a first phase and a frequencyequal to a carrier of the received signal and a first matched filterresponsive to an output of the first multiplier, the second channelincluding a second multiplier responsive to the received signal and asecond reference wave having the carrier frequency and a second phaseorthogonal to the first phase and a second matched filter responsive toan output of the second multiplier, the first and second matched filterssupplying signals to the sampling means, the first and second matchedfilters being included in said filter means.
 25. The receiving equipmentof claim 24 wherein the combining means includes a third multiplierresponsive to the sampled signal derived by the first channel and areplica of the sampled signal derived by the second channel delayed byMT, where M is an integer, and a fourth multiplier responsive to thesampled signal derived by the second channel and a replica of thesampled signal derived by the first channel delayed by MT, and means forlinearly combining product signals derived by the third and fourthmultipliers.
 26. The receiving equipment of claim 25 further includingmeans for detecting the polarity of an output signal of the means forlinearly combining.
 27. The receiving equipment of claim 26 wherein M isan odd integer greater than one.
 28. The receiving equipment of claim 27wherein M is an odd integer greater than one.
 29. The receivingequipment of claim 28 wherein the filter means is responsive to thesignal received by the receiver, the first channel including a firstmultiplier responsive to an output signal of the filter means and theoutput signal as delayed by MT, where M is an integer, the secondchannel including a second multiplier responsive to the output signaland the output signal as delayed by a linear combination of MT and onequarter cycle of the output signal, the sampling means being responsiveto orthogonal components derived from said first and second multipliers.30. The receiving equipment of claim 29 wherein the means for combiningincludes: a third multiplier responsive to a sampled signal derived bysaid sampling means in response to the first multiplier and a firstreference wave having the frequency f₀ MT and a reference phase, wheref₀ is the frequency derived by said filter means, a fourth multiplierresponsive to a sampled signal derived by said sampling means inresponse to the second multiplier and a second reference wave having thefrequency f₀ MT and a phase orthogonally related to the reference phase,and means for linearly combining components derived from said third andfourth multipliers.
 31. The receiving equipment of claim 30 furtherincluding means for detecting the polarity of an output signal of themeans for linearly combining.
 32. Differentially coherent receivingequipment in an MSK transmission system wherein an encoder at atransmitter derives a message having an ordered string of K symbolssuccessively derived with a predetermined periodicity, T, into anordered coded string of K differentially coded symbols, where K is aninteger greater than one, the encoder deriving an MSK continuous phasesignal modulated by a waveform having a string of impulses carrying saidordered coded string, said receiving equipment comprising differentiallycoherent filter means matched with the waveform of the impulses of saidimpulse string, and means cascaded with said filter means forsuppressing intersymbol interference inherent to said MSK typecontinuous phase signal from samples of a string of received impulses.